You're going to need an Xwindows server on your Windows box in order to run graphical Unix apps remotely on the Sun server and have it display on your Windows box. I don't think Tunnelier supports Xwindows tunneling. Take a look at Xming, an Xwindows server for Windows that comes with Putty, an ssh client:
http://sourceforge.net/projects/xming
edit: Glad to see this worked for you. Here's some more explanation on what's happening. X-Windows, the Unix graphical environment is client-server based. IE: it's able to display individual graphical windows on remote systems without full-screen software like VNC or remote desktop. A graphical program in Unix is called the X-Windows client, and the thing that actually does the displaying is called an X-Windows server.
Now, Bitvise Tunnelier is just an ssh client. IE: it only deals with command-line terminal connections. However, the ssh protocol is actually able to tunnel X-Windows over ssh, but you need two things: 1) an X-Windows server running on your desktop (to actually display the app), and 2) an ssh client that supports X-Windows tunneling. Enter Xming, a lightweight X server for windows, and Putty, the ssh client.
So, you were fine ssh-ing in to your Sun box, and typing terminal commands, but Visual SlickEdit is an X-Windows client app. To run that, you needed an X-Windows server. When an X-Windows server is available, it sets the DISPLAY variable on the terminal to tell graphical apps where to display stuff.
One more note: Some of the answers below recommended that you set the DISPLAY variable to the hostname of your Sun box. That might have worked, but it would have displayed the VS windows on the Sun's screen, not your Windows box.